aiqbench analysis
The paper proposes adding small, controlled noise to the unobserved entries of a matrix-completion problem so that the resulting surrogate problem stays within a small error of the true solution and satisfies the restricted isometry property, which enables tensor-based analysis with formal guarantees. It is a theoretical preprint, and its claims that the guarantees hold without sampling-rate or incoherence requirements should be checked against the full proofs and any empirical validation. For ML readers, it is a useful data point on why noise and over-parameterization can help non-convex optimization, though it does not yet show practical gains.
Improving the Optimization Landscape of Matrix Completion with $\epsilon$-close Surrogates
arXiv:2610.41455v1 Announce Type: new Abstract: Non-convex optimization is central to machine learning and AI, yet unlike its convex counterpart, it still lacks a general and principled framework. In this work, we try to highlight the importance of combining noisy perturbations with over-parametrization mathematically under the classic setting of matrix completion (MC). Matrix completion is an important non-convex recovery problem in which only a subset of a matrix's entries is observed, and the remaining entries must be recovered using low-rank constraints. This problem is notoriously difficult because observations are limited. However, we show that by injecting $\epsilon$-level perturbations into the nullspace of the observation mask, we can construct noisy matrix sensing surrogates whose solutions remain $\mathcal{O}(\epsilon)$-close to the ground truth with valid restricted isometry property (RIP) constants. The existence of RIP further enables powerful tensor frameworks to be applied with theoretical guarantees. This approach reflects a broader empirical trend in which stochasticity or noise improves the curvature of the optimization landscape, allowing over-parameterized (a.k.a large) models to better separate signal from noise. Assuming each matrix entry is observed independently, we establish quantitative recovery guarantees without explicit requirements on sampling rate or incoherence, and discuss how such assumptions can further strengthen our results.
Abstract:Non-convex optimization is central to machine learning and AI, yet unlike its convex counterpart, it still lacks a general and principled framework. In this work, we try to highlight the importance of combining noisy perturbations with over-parametrization mathematically under the classic setting of matrix completion (MC). Matrix completion is an important non-convex recovery problem in which only a subset of a matrix's entries is observed, and the remaining entries must be recovered using low-rank constraints. This problem is notoriously difficult because observations are limited. However, we show that by injecting $\epsilon$-level perturbations into the nullspace of the observation mask, we can construct noisy matrix sensing surrogates whose solutions remain $\mathcal{O}(\epsilon)$-close to the ground truth with valid restricted isometry property (RIP) constants. The existence of RIP further enables powerful tensor frameworks to be applied with theoretical guarantees. This approach reflects a broader empirical trend in which stochasticity or noise improves the curvature of the optimization landscape, allowing over-parameterized (a.k.a large) models to better separate signal from noise. Assuming each matrix entry is observed independently, we establish quantitative recovery guarantees without explicit requirements on sampling rate or incoherence, and discuss how such assumptions can further strengthen our results.
Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2610.41455 [cs.LG]
(or arXiv:2610.41455v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.41455 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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